Wavelets [1]}, [2]}, [3]}, [4]}, [5]}, [6]} have been thoroughly investigated for over two decades. Their efficiency and success in analyzing functions defined on the real line have led to their generalization for functions defined on higher dimensional spaces such as the sphere [7]}, [8]}, [9]} or other manifolds [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}. The development of wavelets on manifolds and the multi-scale feature of graphs and data on graphs led to the extension of wavelets construction to the discrete space setting in recent years. These wavelet-type constructions include wavelets on unweighted graphs [17]}, lifting wavelets [18]},
diffusion wavelets [11]}, [12]}, [21]}, diffusion polynomial frames [10]}, spectral graph wavelets [23]}, [24]}, [25]}, Haar-like wavelets [26]}, [27]}, [28]}, [29]}, average interpolating wavelets  [30]},
graph wavelets via deep learning [31]}, multi-scale pyramid transform [32]}, tight wavelet frames on graphs [15]}, [34]} and intertwining wavelets on graphs via random forests  [35]}. The wavelet constructions mentioned above were all developed on undirected graphs. In the present work, we intend to construct a harmonic analysis on directed graphs. The first stage of the proposed harmonic analysis is the development of a Fourier analysis.
